Chapter 1, Problem 26SE

Suppose that an invertible matrix A is partitioned as

$A=\left[\begin{array}{l}{A}_{11}{A}_{12}\\ A_{21}{A}_{22}\end{array}\right]$

Show that

$A^{-1}=\left[\begin{array}{l}{B}_{11}{B}_{12}\\ B_{21}{B}_{22}\end{array}\right]$

Where

$B_{11}={(A_{11}-A_{12}{A}_{22}^{-1}{A}_{22})}^{-1}$, $B_{12}=-B_{11}-A_{12}{A}_{22}^{-1}$

$B_{21}=-A_{22}^{-1}{A}_{22}{B}_{11}$, $B_{22}={(A_{22}-A_{21}{A}_{11}^{-1}{A}_{12})}^{-1}$

provided all the inverses in these formulas exist.